- Knowledge and ability to understand
At the end of the course the student must demonstrate a good
knowledge of the theory of independence, of the main stochastic
processes, both in discrete time and in continuous time, and in the basic
notions of stochastic calculus. The student must be able to recognize the
reasoning and the techniques of the disciplines covered by the course.
- Ability to apply knowledge and understanding
At the end of the course the student must know how to model some
types of problems of physics and mathematical finance through
applications of probability calculus.
- Autonomy of judgment
The student must know how to recognize independently which types of
models must be applied to solve advanced probabilities.
- Communication skills
The student must know how to express with language properties about
- Learning skills
The student must know how to read and learn probability concepts from
advanced university texts
Main results of the probability.
Independence theory. Conditional mean. Brownian motion. Martingales.
Markov chains. Markov processes. Poisson processes. Stochastic
Classroom lectures, classworks, homeworks.
• Monotone class theorem.
• Independence theory.
• Conditional mean. Conditional variance.
• Stochastic processes. Stopping times.
• M-dimensional Fourier transform. Gaussian vectors.
• Brownian motion. Paul Lévy theorem. Iterated logarithm law. Mdimensional Brownian motion.
• Discrete martingales. Martingales convergence theorem. Uniform
integrability and L^1-convergence. Large numbers strong law.
• Markov chains. Recurrence, positive recurrence, nul recurrence,
transience. Irriducibility, aperiodicity. Invariant measures. Convergence
• Markov processes.
• Poisson processes.
• Stochastic integration.
Oral final examination.
The goal is to introduce the stochastic calculus.
Equazioni differenziali stocastiche e applicazioni. Paolo Baldi. Pitagora
Editrice. Bologna, 2000, II ed.